Translations: generalizing relative expressiveness between logics
There is a strong demand for precise means for the comparison of logics in terms of expressiveness both from theoretical and from application areas. The aim of this paper is to propose a sufficiently general and reasonable formal criterion for expressiveness, so as to apply not only to model-theoretic logics, but also to Tarskian and proof-theoretic logics. For model-theoretic logics there is a standard framework of relative expressiveness, based on the capacity of characterizing structures, and a straightforward formal criterion issuing from it. The problem is that it only allows the comparison of those logics defined within the same class of models. The urge for a broader framework of expressiveness is not new. Nevertheless, the enterprise is complex and a reasonable model-theoretic formal criterion is still wanting. Recently there appeared two criteria in this wider framework, one from García-Matos & Väänänen and other from L. Kuijer. We argue that they are not adequate. Their limitations are analysed and we propose to move to an even broader framework lacking model-theoretic notions, which we call “translational expressiveness”. There is already a criterion in this later framework by Mossakowski et al., however it turned out to be too lax. We propose some adequacy criteria for expressiveness and a formal criterion of translational expressiveness complying with them is given.
- 1 Introduction
- 2 Multi-class expressiveness
3 Translational expressiveness: obtaining a still wider notion of expressiveness
- 3.3 Mossakowski et. al.’s approach
- 3.7 Adequacy criteria for expressiveness
- 3.17 : a sufficient condition for expressiveness
- 3.27 Corroborating : the structure preserving translations
- 4 Conclusions
It is very common for those who work with logic to make comparisons such as “the logic is more expressive than ”, “ is stronger than ”, “ is included in ”, “ can be reduced to ”, etc. Such assertions are often made on imprecise grounds and, though possibly being non-ambiguous and non-problematic, the lack of clarity around the usage of these concepts can generate terminological confusion across the literature (e.g. [Hum05]) and harden the comparison of formal results.
In the literature, the notion of logic inclusion or sub-logic (these terms will be used interchangeably here) is pretty much linked with language and axiomatic extensions, which on their turn are linked with “strength”, that is, the capacity of proving theorems or having valid formulas. Now the concept of sub-logic is sometimes associated with strength and sometimes associated with expressiveness, and sometimes with both (e.g. in [Béz99]), which is known to be the case of paradoxes [MDT09]. Three kinds of systems are relevant here: model-theoretic logics, Tarskian and proof-theoretic logics, they will now be briefly defined. A logic is called model-theoretic if it is defined semantically and presented as a sequence (), where is a set of formulas, is a class of models and is a satisfaction relation on . A logic is Tarskian if it is defined as (), where is a consequence relation on (possibly multi-consequence). Finally, is a proof-theoretic logic if it is defined as , where is a set of inference rules.
In model-theoretic logics there is a straightforward approach to expressiveness that is also reasonably taken as a definition of logic inclusion: a logic is at least as expressive/includes if every class of structures characterizable in is also characterizable in (see e.g. [Lin74, p. 129] and [BF85]). This naturally only holds for logics defined within the same class of structures. If one wants also to compare logics defined within different classes of structures, then it does not seem adequate to use the concept of sub-logic, as we shall see below. It is better to use the concept of expressiveness.
There is no straightforward approach to expressiveness for Tarskian and proof-theoretic logics (TPL, for short). As for sub-logic, in TPL it is also linked with language and axiomatic extensions. However, we can often see “sub-logic” relations taken in a wider sense, i.e. when, for two given logics and , it happens that is not a language/axiomatic extension of , but there is a certain mapping of -formulas into -formulas respecting the consequence relation. These cases are normally interpreted as saying that is included/embeddable/reconstructible/interpretable/can be simulated in . We propose to call these as expressiveness relations whenever they can be seen as modeling the following intuition
For every -sentence , there is an -sentence with the same meaning.
This same intuitive explanation of expressiveness holds for model-theoretic logics, and is used as a basis for formal criteria therein (e.g. [BF85, p. 42]). Thus we can have a reasonably homogeneous concept for comparing logics: that of expressiveness. We shall reserve the term “sub-logic” just when there are axiomatic or language extensions, and we shall not use the term “strength” because it is ambiguous between expressive and deductive strength.
A precise definition for the notion of relative expressiveness for model-theoretic logics was given already in the 1970s (e.g. in [Lin74] and [Bar74]). As we said, this definition is based on the capacity of characterizing structures and underlies each of the so-called Lindström-type theorems,
Single-class expressiveness Considering model-theoretic logics defined within the same class of structures, the above intuition can be captured easily since there is a common ground where sentences can be compared. This common ground is easily achieved by defining the meaning of a sentence in a logic as (, for short). Thus we call this framework single-class expressiveness. Since every sentence in is mapped to a sentence in having the same meaning, this framework of expressiveness can be seen as consisting of certain formula-mappings between model-theoretic logics. A formal definition for it is then straightforward. Let be a signature and let and be model-theoretic logics.
Definition 1.1 ().
is at least as expressive as () if and only if (iff, for short) for every sentence there is a sentence such that .
Notice that here the class of models is the same for both and , and share the same non-logical symbols. The above definition can be paraphrased in terms of elementary classes:
Despite being the basis for many important results, is very limited. It is not only restricted to model-theoretic logics, but it requires the classes of structures being compared to share the same signature. As a consequence, it only allows the comparison of logics defined within the same class of structures. The urge for a broader definition is not new.
Even among those expressiveness results using , we can notice some flexibility in its application. One such example appears in [AFFM11], where the definition of above is given, but afterwards (p. 307) it is informally relaxed in order to allow changes of signature, thus the proper definition being used appears to be the one based on projective classes (). The problem is that elsewhere we get different results depending on whether we use or , as Shapiro showed [Sha91, p. 232]: and , but and .
Remaining within model-theoretic logics, a wider framework —let us call it multi-class— would comprise besides formula-mappings also structure-mappings, thus allowing structures of one logic to be mapped to structures of the other. This would enable the comparison of logics defined within different classes of structures. Recently there appeared two formal definitions of multi-class expressiveness, to wit [GMV07] and [Kui14]. In the sequence we will present them and argue that they are not adequate.
There have been also early claims outside abstract model-theory relating logics in the sense of (E) above, but no explicit definitions of the main concepts involved were given. Gödel used his result on the interpretation of classical into intuitionistic logic to infer that, contrary to the appearances, it is classical logic that is contained in intuitionistic logic [Göd01, p. 295]. Since then, there followed many results of interpretations, embeddings, reconstructions, simulations, etc. among Tarskian and proof-theoretic logics. Such results have often been used to justify some statement of inclusion or relative expressiveness between the logics at issue.
As opposed to the case of model-theoretic logics, until recently there was no attempt to give a precise definition of relative expressiveness in this framework. To the best of our knowledge, Mossakowski et al. [MDT09] were the first to give an explicit formal definition of translational expressiveness for logics, that is, an expressiveness relation based on the existence of certain kinds of formula-mappings. We will expose their definition and show that it is still not adequate. Then, some adequacy criteria for expressiveness are proposed and a formal criterion for translational expressiveness is given.
Structure of the paper
This paper presents the following panorama on relative expressiveness between logics:
Relative expressiveness between logics (intuitive concept as given by (E))
Adequacy criteria for expressiveness
Approaches to (*) hopefully satisfying (a)
formal proposals: ,
formal proposals: ,
formal proposals: Mossakowski et al.’s and .
In the framework of multi-class expressiveness will be presented and two formal criteria will be analysed, one from [GMV07] () and other from [Kui14] (). We argue that, using the intuitive explanation of expressiveness given above, there are counterexamples to both. In the sequence, we investigate what is wrong with them and propose that moving to an even wider framework, encompassing a greater range of logics and lacking structure-mappings, might be promising.
In we present Mossakowski et al.’s formal criterion for translational expressiveness and show that, due to a result of [Jeř12], it is still not adequate. Then, some basic adequacy criteria for expressiveness will be proposed. In the sequence we analyse some formal conditions related to translations already appearing in the literature and investigate whether they satisfy the adequacy criteria. Finally, a formal sufficient condition for translational expressiveness () is proposed. We will argue that satisfies the criteria and is materially adequate.
2 Multi-class expressiveness
2.1 M. García-Matos and J. Väänänen on sub-logic
García-Matos and Väänänen gave a multi-class definition of sub-logic. Their definition is similar to one given in [Mes89] but is laxer.
A logic is a sub-logic of (in symbols ) if there are a sentence and functions , such that:
For every exists a such that and
For every and for every , if , then ( iff )
Thus, if the class of structures of a logic is richer than the class of structures of a logic , one could still allow a comparison between and , by restricting to the translatable structures, i.e. those which satisfy some condition and then use a function to translate this reduced class of -structures into -structures.
A problem with
Let be a trivial propositional logic in some given signature, and let () be the set of is truth tables together with a valuation. Let be any logic that has at least one valid sentence and let the formula of the definition above be such . Define the following mappings
. For every , .
. For every , .
Then it is easily seen that both items (a) and (b) above are satisfied.
Thus, according to this definition of sub-logic, every logic containing at least one valid formula has a trivial sub-logic. If we think on the usual meaning given to “sub-logic”, this not plausible at all, since the logic could be non-trivial and might even lack a trivializing particle, so how come it could have a trivial sub-logic?
It is not enough to require that the mapping be injective. Using an idea of [CCD09, p. 14], take for target logic any that has a denumerable number of valid formulas and define the mapping from the formulas of the trivial logic to -formulas as . Still we have that has a trivial sub-logic, once more, may be any logic with a denumerable number of validities, also lacking a trivializing particle.
Naturally, the usual senses of logic inclusion, that is, through language or axiomatic extensions do not apply here. The only way to make sense of this is to interpret the above cases as saying that a trivial logic can be simulated in any logic containing at least one validity. This capacity of simulating a logic is an expressive capacity, therefore the definition above is better seen as a definition of expressiveness. Yet, as an expressiveness relation, it is noteworthy that no restriction on the translation functions and are imposed, so one may wonder whether the definition over-generates.
We are not in position to settle definitively this question. However we will give a plausibility argument to the effect that we should impose stricter conditions on model- and formula-mappings, since there is a natural and reasonable extension of the above definition that indeed over-generates. Though not, strictly speaking, a counter-example, the case to be presented below shall give evidence that there is an intrinsic problem with the above proposal for multi-class expressiveness.
As we said, the sentence on the above definition of is intended to cut -structures that are meaningless from the point of view of . Apparently, it would do no harm to the idea behind to allow to be a recursive set of sentences, as it is normally done in works dealing with translations of logics and conversion of structures (e.g. [Man96, p. 270]). This would be useful if the logics at issue have no conjunction, so that could be a finite set of sentences; or if the low expressive power of the logics and makes that the -structures to be reduced into -structures be only characterizable through an infinite but recursive set of -sentences. This happens in the case of many-sorted logic () and . If is not allowed to be an infinite set of sentences, then would not be a sub-logic, in the above sense, of , which is implausible. Though the conversion of -structures into -structures is mentioned [GMV07, p. 23], the case of a given -signature containing infinitely-many unary symbols is not considered. To convert -structures into -structures then one needs to make sure that unary predicates to be converted to many-sorted domains are non-empty. This would only be accomplished by setting [Man96, p. 260].
However, if one allows such modification another implausible situation occurs. Consider the classical propositional logic () and a propositional logic , defined by Béziau [Béz99]. shares all the definitions of the classical propositional connectives, except for negation, where it has only one “half” of its clause: for a -model and formula , if , then ; the converse direction does not hold.
Béziau shows that there is a translation from into . Below we will give Mossakowski et al.’s presentation of it, which includes also a model translation [MDT09, p. 107]. Given an n-ary connective , a translation is literal for if ; for an atomic formula , is literal when . Define the mapping ( as follows:
literal for and atomic formulas;
where comprises the truth-tables for each connective and a valuation on the propositional variables. Notice that takes a -model, keeps the valuation and replaces the truth-tables for the corresponding ones.
Then we have that
Theorem 2.3 (Mossakowski et al.).
if and only if .
The model mapping is surjective, so that it obeys (a) above.
Now Mossakowski et al. (ibid, p. 100) define a mapping also from to using an auxiliary set of formulas constructed out of -formulas.
Define the mapping as follows:
For every , , where is a propositional variable.
Define as the following set of formulas, for :
The purpose of is to encode the semantics of into the propositional variables , since every -formula is translated into one of such , in a -model satisfying the valuation of the propositional variables is forced to respect the semantics of . For example, in , if , then it holds that , but the converse direction does not hold. This is simulated in the -models satisfying by the fourth clause above: if , then which implies that . But, as in , it does not hold that if , then .
Now define the model-translation :
Let be a -model satisfying . Then is defined as follows:
For every -formula , iff .
is also surjective (so it obeys (a) in the criterion for sub-logic above). Then we have that
Theorem 2.4 (Mossakowski et al.).
iff and .
Therefore, by the above results and according to the extended definition of sub-logic, we would have that and are one sub-logic of another, which is not plausible. is not a sub-logic of in the sense of language/axiomatic extension. Neither they are expressively equivalent, using above, since the “half-negation” present in is not available in .
The problem is that the translation from to uses a trick to sneak in the semantics of into . Restricting the -models that satisfy , one simulates the behaviour of -formulas in the propositional variables and sustain such behaviour through the model-translation.
The modified version of , allowing to be a recursive set of sentences looks at least as “natural” as the original one. Even considering the original definition 2.2 we can see that there is something wrong with it, in not requiring any kind of preservation of the structure of formulas e.g. by forcing to be inductively defined through the formation of formulas. Then one may conjecture that, among more expressive logics, there be translations () where maps entire formulas to propositional variables and, with a sentence restricting the target structures, is able to mimic the semantic behavior of . Then it is very doubtful that the obtained would have the same meaning as .
Thus, we think we have good reasons to consider that García-Matos and Väänänen’s definition of sub-logic is not adequate. It would certainly be better to use a stronger notion of translation, paying attention to the structure of formulas. Only then the meaning of the target-formulas could be said to match the meaning of the source-formulas. Below we will see that a development along this line appeared in the literature. Nevertheless, there is still a structure-attentive translation that “cheats” similarly as the one above, mimicking the semantics of one logic into the other.
2.5 L. Kuijer on multi-class expressiveness
In his doctorate thesis [Kui14] Kuijer studies the expressiveness of various logics of knowledge and action, these logics are taken in the model-theoretic sense. He notices that there are some results relating logics similarly as in single-class expressiveness.
The purpose is to investigate features shared by all the results and construct a criterion, to be called “”, based on these features. Similarly with the work of García-Matos and Väänänen exposed above, these prototypes involve translations of sentences and translations of structures. So a translation from to is a pair (), with and or , such that satisfies some given conditions.
A first plausible condition is that () must preserve and respect truth:
Definition 2.6 (Truth preserving).
A translation with and is truth preserving if, for every and
if and only if .
Then a tentative definition of could be
is at least as iff there is a that is truth preserving.
The problem is that the requirement of truth preservation is very weak, indeed there are several trivial truth-preserving translations among almost every logic. Kuijer gives the following example [Kui14, p. 88].
A trivial translation
Let be any logic on possible world semantics such that is countable and let be a logic where is a countable set of propositional variables but with no connectives and where is a class of models with possible worlds. Thus, every is a set of possible worlds with a valuation.
Define a truth-preserving translation () from to in the following way: map every to a propositional variable , maps a model to a model taking the set of possible worlds of and removing every other structure, and with the following valuation . Then clearly, by definition, is a truth preserving translation.
Since in the above example is an arbitrary logic on possible world models, if truth preservation were the only condition for multi-class expressiveness, would be at least as expressive as , which is absurd, given that has scarce expressive means. Nevertheless, truth-preservation is clearly a necessary condition. Thus, one must find other features a translation must satisfy in order to serve as a formal elucidation of the notion of multi-class expressiveness.
Another immediate criterion that comes to mind in order to avoid the trivial translations is to require the preservation of validities and entailment relations. However, some of the chosen prototypical translations do not preserve validity and some do not preserve entailment. Since the idea was to capture the essential features shared by all prototypical translations in , none of these can be imposed as a necessary condition.
Kuijer then goes through a number of tentative criteria, e.g. preservation of atomic formulas, of sub-formulas, etc., and shows that they are either too lax or too restrictive. Among the lax criteria, that is, the ones that are satisfied by some trivial translation, is one that Kuijer considers nonetheless important, the criterion of being model based:
Definition 2.7 (Model based).
A translation is model based if there are two functions such that, for all , we have that .
A model based translation would force to preserve some structure of and prevent that the pointed models and be translated to completely unrelated models.
Finally, the condition that apparently divides the good from bad translations and gives a reasonable notion of multi-class expressiveness is the criterion of being finitely generated. For the sake of simplicity, some aspects of the definition below are not completely formalized.
Definition 2.8 (Finitely Generated).
Let and be such that is generated by a set of propositional variables and a finite set of connectives, for . Let , then a translation is finitely generated if can be inductively defined by a finite number of clauses of the form
where is an -sentence constructed out of and possibly containing , for ; and where is the range of the , e.g. if a given is to be replaced by a formula or only by an atomic formula.
The set contains the special propositional variables to be used in the translation clauses, for which one can substitute formulas. An example of such a translation clause is: for ; and for .
The idea is that (ibid, p. 110) it is the fact of being inductively defined and thus respecting (some) of the structure of the formulas that sets the finitely generated translations apart from the trivial translations. Thus Kuijer concludes that the truth-preserving translations giving rise to an expressiveness relation could be characterized as the ones being finitely generated and model-based. Therefore, the final criterion given for multi-class expressiveness is (ibid, p. 111)
Definition 2.9 (Expressiveness).
Let and be such that is generated by a set of propositional variables and a finite set of connectives for .
Then is at least as as iff there is a translation from to that is model based, finitely generated and truth preserving.
A problem with
Kuijer had no pretensions that his multi-class definition were to be the generalization of expressiveness as given by the single-class framework. The aim was to find only a “reasonable generalization” (ibid, p. 83). While keeping this in mind, we would like to argue that his proposal is still not good enough as a criterion for multi-class expressiveness. This is because one can find a pair of logics , such that is intuitively more expressive than , although is at least as as .
The logics at issue are Epstein’s relatedness logic () [Eps13, p. 80] and classical propositional logic (). The logic besides the truth-functional connectives, has a relevant implication “”, which is the reason it is intuitively more expressive than , which lack such a connective. The referred translation would imply that is at least as as .
Despite the circumscribed character of Kuijer’s criterion, we think that a reasonable generalization of single-class expressiveness should be able to deal with a reasonable amount of logics, not only with a handful of them. Particularly when the logics at issue are in the literature, and have not been constructed in an ad-hoc fashion just to give a counter-example. Finally, there is nothing specific about the logics appearing in the counter-example, so it is quite possible that there are also modal counter-examples.
Epstein presents with the connectives . The first two are defined as usual and the underlying idea for interpreting the relevant implication symbol “” is as follows. It holds that whenever materially implies and both are subject-matter related to each other through a relation defined on all propositional variables. Specifically, for propositional variables and -sentences and , holds if and only if for some occurring in , and occurring in , it holds that . Thus, the truth table for “” is the one for material implication with an additional column for , so that if holds and is true, then is true; else, if does not hold, then is false.
Let be a signature for . An -model () is formed by the truth-tables for , a symmetric and reflexive relation on -formulas and a valuation . For propositional variables , let . Let be defined on (note we use here to emphasize that it is a material implication).
We will see below that there is a truth-preserving, model-based and finitely generated translation . The mapping is defined as follows:
literal for and atomic formulas.
Here the basic idea for the translation of comes from the definition of “”: materially implies and both formulas are related through . As the translation is defined inductively through the formation of formulas by a finite number of clauses, it is finitely generated.
Now, from an -model (), one easily defines a transformation from -models to -models. Let , where, for take all the truth-tables in , excluding the one for . Define as follows (adapted from [Eps13, p. 300]):
Clearly is model-based.
Both and satisfy a semantic deduction theorem (ibid, p. 299). To prove that is truth-preserving, one has to prove only that, for an arbitrary -model , it holds that
Theorem 2.10 (adapted from Epstein).
if and only if .
is at least as as .
The main question now is: does show that is at least as expressive as ? We do not think it is reasonable to say so, since the extra expressiveness brought about by the implication connective in is only by a trick mimicked in . Independently of the model-translation to give the intended truth values for the “relevance-mimicking” variables , it is not possible to have a relevant conditional in , by say, adjoining to a conditional such variables . To do so, would require too much for the intended meaning of such variables. Surely this would not augment the expressive power of the propositional logic, as it concerns only an interpretation of propositional variables, and intuitively, specific interpretations of propositional variables do not influence the expressiveness of a logic.
Anyway, the model-mappings are not essential for these translations using indexed variables, they only facilitate their definition. An early example was given by Richard Statman in [Sta79] where a translation of into its implicational fragment is presented. There, the conjunctions are mapped to implications containing , among formulas of the sort , , etc. Here the situation is entirely different since the proof-theoretic behaviour of individual conjunctions are encoded in specific variables using implicational axioms.
Coming back to Kuijer’s criterion, we argued above that it is not enough to give an intuitively adequate account of expressiveness. If the model mapping were not from the source logic to the target logic but vice-versa, then there would not be such truth preserving mappings from to , as there would be no way to construct the relatedness predicate out of a -model. Kuijer discarded such a definition of the model mappings since it implies that any truth-preserving translation is also validity preserving,
Let us analyse a possible strengthening on the formula translation. We will not give a detailed analysis of features of translations since it suffices to notice that Epstein’s translation preserves completely the structure of the formulas, except for . For this case, additional propositional variables must be introduced to bear the intended meaning of (variables whose interpretation in is sustained by the model translation.) If one required that be compositional, that is, every -ary connective of the source logic is translated by a schema of the target logic, then the above translation would not pass the test. This is because is translated through the schema , and by the schema . If the translation were compositional, dealing with the same connective, the same translation schema would be used.
The problem of adopting this criterion is that it implies that the connectives be translated one at a time, and again some of the paradigmatic translations selected by Kuijer takes into consideration sequences of connectives, so they would not satisfy it.
Therefore, to prevent translations such as those above from passing the test for multi-class, one would have to use a criterion for that is stronger than being finitely generated, but weaker than being compositional. Nevertheless, the enterprise of placing restrictions on the formula translations alone seems not to be promising, as the model-translations play a major role in the counter-examples presented above. On the other hand, placing also restrictions on model-translations and making them fit with the restrictions on formula-translations is a very complex enterprise, and there may be better alternatives.
Given this situation, we would like to suggest a change of perspective as regards relative expressiveness between logics. Below, some comments will be made regarding the nature of the notion of expressiveness and its relation with the concept of logical system it applies to.
2.12 Single-class expressiveness vs multi-class expressiveness vs translational expressiveness
Now we would like to make some remarks on the study of the relation of expressiveness between logics. As we commented before, in the single-class framework it is very simple to define relative expressiveness, since there is a common ground, the structures, where one can compare whether the sentences have the same meaning. Now consider the multi-class framework, if and are defined on different classes of structures, how would we know whether an -sentence and an -sentence have the same meaning? After all, in this case it trivially holds that .
As we saw, for this task new tools are needed: a model-mapping or ;
Thus, now the weight goes on the notion of translation . As we saw in the examples presented above, for () in , and () in , the task of establishing the congruence between the pairs (, ) and (, ) by means of translations is very difficult. Basing it on satisfaction is far away from being sufficient, since we can easily devise translation functions such that satisfies iff satisfies .
On the other hand, imposing conditions on is a complex enterprise, because either it under-generates or, by a little breach, it over-generates. Moreover, the need to have model-mappings besides formula-mappings may open up a back door to undesirable translations, to see it, consider again the examples offered against García-Matos & Väänänen’s and Kuijer’s approaches. All of them use some “trick” in the formula-translation function and sustain it through the model-translation. Then it is of little help to place structural restrictions on formula-translations, as did Kuijer. He also tried placing restrictions on model-translations, but it did not help either.
Therefore, it might be more promising to move to a wider framework of relative expressiveness, dispensing with the semantic notions altogether. In this framework, to be called “translational expressiveness”, we would then concentrate the investigations on the conditions on formula translations. The aim is to find the set of conditions that better preserve/respect the theoremhood/consequence relation and the structure of formulas of each logic. This way a reasonable formal criterion of expressiveness for Tarskian and proof-theoretic logics (TPL, for short) would be obtained, and a bigger range of logics would be comparable. Finally, these advantages would arguably come at no cost, since this wider enterprise would not be more difficult than multi-class expressiveness.
The big difference between the approaches of expressiveness is not in the division between expressiveness for model-theoretic logics and for TPL, but in the division, in model-theoretic logics, of expressiveness within the same and within different classes of structures. Naturally the most direct concepts of expressiveness are linked with the capacity of characterizing structures, but this only applies when comparing the same class of structures.
If one allows translations between structures, such capacity is no longer at issue. Once we depart from the safe harbour of a single class of structures for comparing logics, then all bets are off. Multi-class expressiveness does not guarantee a firmer grasp of the intuitive concept of expressiveness anymore than translational expressiveness. Since the move to a wider framework might not only free us from problems inherent to multi-class expressiveness, but also allow a bigger range of comparison of logics, then the prospects for the enterprise are better.
As we said in the introduction, people have been using informally some concepts of translational expressiveness between logics. However, as opposed to what happens with model-theoretic logics, to the best of our knowledge, in the literature there is only one explicit and formal criterion in this framework, that of [MDT09]. In the next section, their proposal will be analysed and we will show that it is not adequate. We shall then propose some adequacy criteria for expressiveness and a formal criterion in the framework of translational expressiveness will be given. We then argue that the criterion satisfies the adequacy criteria.
3 Translational expressiveness: obtaining a still wider notion of expressiveness
In this section we will deal with logics in the Tarskian and proof-theoretic sense. We also mention logics taken as a closed set of theorems/validities, to be called simply “formula logics”. Let and be logics, be a set of -formulas and a translation mapping -formulas into -formulas in such a way that for each -formula :
if and only if .
In this case is translatable into with respect to theoremhood.
If it is the case that
if and only if
Definition 3.1 (Conservative translation).
A conservative translation is a translation with respect to derivability.
Whenever we want to refer indistinctly to translations with respect to theoremhood or conservative translations, the term back-and-forth will be employed.
Definition 3.2 (Back-and-forth translation).
A translation is back-and-forth if it is either a theoremhood preserving or a conservative translation.
3.3 Mossakowski et. al.’s approach
As far as we know, Mossakowski et al. [MDT09] proposed the first explicit criterion for the concept of sub-logic and expressiveness in the framework of translational expressiveness:
Definition 3.4 (Sub-logic).
is a sub-logic of if and only if there is an injective conservative translation from to ;
Definition 3.5 (Expressiveness).
is at most as expressive as iff there is a conservative translation .
The authors do not explain why sub-logic requires injective conservative mappings while expressiveness does not. Anyway, we will see that these criteria for sub-logic and expressiveness via conservative mappings do not work.
The conception that conservative translations could give rise to a notion of expressiveness and also a notion of logic inclusion has been supported more than once. For example, in [Con05, p. 233] it is said that the existence of a conservative translation (maybe injective or bijective) would give rise to some kind of logic inclusion between Tarskian logics.
Unfortunately, conservative translations will not make a reasonable concept neither of sub-logic nor of expressiveness. Due to a result of Jeřábek [Jeř12], explaining expressiveness and sub-logic through conservative translations would make include and be at least as expressive as many familiar logical systems, e.g. first-order logic. He proved the following result (ibid, p. 668), where for a logic , a translation is most general whenever it is equivalent to a substitution instance of every other translation of to .
Theorem 3.6 (Jeřábek).
For every finitary deductive system over a countable set of formulas , there exists a conservative most general translation . If is decidable, then is computable.
The defined mapping is injective.
The author criticizes the notion of conservative translation for not requiring the preservation of neither the structure of the formulas nor the properties of the source logic [Jeř12, p. 666]. Thus, it must be strengthened in order to serve for an expressiveness measure. This could be done in a simpler way by requiring injective, surjective or bijective mappings. As Jeřábek’s mapping is injective, only requiring injectiveness will not do. As a matter of fact, it seems that already requiring injectiveness one is overshooting the mark. Since in this way would not be as expressive as . Any mapping would have to map both -sentences and to the same -sentence , so it would not be injective.
Other kinds of strengthening hinted by Jeřábek’s (ibid) are:
force the mappings to preserve more structure of the source logic sentences in the target logic;
force the mappings to preserve more properties of the source logic.
The adequacy criteria for expressiveness to be given below will require to some extent (1) and (2).
3.7 Adequacy criteria for expressiveness
As we saw above, Mossakowski et al. [MDT09] gave a proposal for a wide notion of expressiveness: by means of the existence of conservative translations. Due to Jeřábek’s results on the ubiquity on this kind of translation, their definition is not adequate. Maybe we should step back and think about some adequacy criteria every approach to expressiveness ought to accomplish.
The intuitive explanation for expressiveness given in the beginning elucidates relative expressiveness in terms of a certain congruence of meanings. It appears already in a more direct form in Wójcicki’s Theory of Logical Calculi [Wój88, p. 67], and we place it as the first adequacy criterion
[Adequacy Criterion 1] is at least as expressive as only if everything that can be said in terms of the connectives of can also be said in terms of the connectives of .
Here, for “being said in terms of the connectives” there can be stricter interpretations (as proposed by Wójcicki, Humberstone, Epstein) and wider interpretations (as proposed by Mossakowski et al. and us), to be developed below.
There are some meta-properties of logics that are intuitively known to limit or increase expressiveness. Thus, the presence/absence of such properties can be used to test whether there can be or not an expressiveness relation between the given logics. A first one coming to mind is that nothing can be expressed in a trivial logic, so it cannot be more expressive than any logic. Another one has to do with the relation between expressiveness and computational complexity. This relation has even been stated as the “Golden Rule of Logic” by van Benthem in [vB06, p. 119], where he says “gains in expressive power are lost in higher complexity”. Nevertheless, the “Golden Rule” is not quite useful here, since we know that in general neither a low expressiveness means low complexity,
Nevertheless the complexity levels of decidability/undecidability can be useful for expressiveness comparisons: if a logic is decidable, then it cannot describe Turing machines, Post’s normal systems, or semi-Thue systems. Therefore, a decidable logic cannot be more expressive than an undecidable logic , otherwise, would not be decidable!
The third meta-property that could be useful when evaluating expressiveness relations (except, naturally, when dealing with formula-logics) is the deduction theorem. Though involved in many formulation issues, as we shall see, a logic has a deduction theorem when it has the capacity to express in the object language its deductibility relation. Thus, other things being equal, a logic having this capability is intuitively more expressive than another one lacking it. Therefore, it is desirable that an expressiveness relation carries with it the deduction theorem, so that (a) below apparently should hold
if is more expressive than , and has a deduction
theorem, then so does .
We have some issues here. Being formulation sensitive, it is complicated to define in which circumstances the existence of a deduction theorem for a logic implies its existence in another logic, whenever there is an expressiveness relation between them. For example, a less expressive logic might have the standard deduction theorem,
Cases like these constrain us to limit the role of the deduction theorem in expressiveness relations, admitting wider formulations of it. Thus we are forced to adapt (a) accordingly so as to be able to take into account such phenomena. Finally, we have the meta-property related adequacy criterion.
[Adequacy Criterion 2] It cannot hold that be more expressive than when
is non trivial and is trivial;
is undecidable and is decidable;
satisfies the standard deduction theorem and the language fragment of purportedly as expressive as does not satisfy (not even) the general deduction theorem;
The last criterion reflects the intuition that expressiveness is a transitive relation and there are logics that are more expressive than others.
[Adequacy Criterion 3] (Taken from [Kui14]) The expressiveness relation should be a non-trivial pre-order, that is, it should be a transitive and reflexive relation, and there must be some pair of logics and such that is not at least as expressive as .
We now analyse with greater detail the criteria 1 and 2.
Criterion 1- on “whatever can be said in terms of the connectives”
We can understand this criterion as saying “every connective of is definable in ”. But the usual notion of definability is either treated within the same logic, or between different logics within the same class of structures. As we intend to deal with translations between logics, the usual notion of definability is too rigid. We must give a broader reading of the criterion 1 in order to understand it as imposing an intuitive restriction on translations between logics. Thus the idea is to impose restrictions on translations so that
satisfies only if, intuitively, everything that can be said in terms of the connectives of can also be said in terms of the connectives of ; let us say in shorter terms that this happens only if the connectives of are generally preserved in .
In the sequence some candidates for such are listed, the back-and-forth condition was given before.
Definition 3.8 (Compositional).
A translation is compositional whenever for every -ary connective of there is an -formula such that .
Definition 3.9 (Grammatical).
A grammatical translation is a back-and-forth compositional translation such that, for a sentence , may contain no other formulas other than the ones appearing in , where appears in (thus, no parameters are allowed).
Definition 3.10 (Definitional).
A definitional translation is a grammatical translation for which for every atomic .
We have four proposals for filling the above list of restrictions. All of them require basically two conditions, taking as the back-and-forth condition. In decreasing order of strictness, there is divergence in taking as a
definitional translation (Wójcicki and Humberstone),
grammatical translation (Epstein and apparently Koslow),
general-recursive translation (to be defined below),
surjective conservative translation (Mossakowski et al.).
Humberstone [Hum05], recalling Wójciki’s definitional translations and intuitions about expressiveness, guessed that if there is a definitional translation between and , then all connectives in are preserved in .
For Epstein [Eps13, p. 302], a grammatical translation is a homomorphism between languages and thus it yields a translation of the connectives. The justification is that such translations are only possible when for each connective in the source logic, there corresponds a specific structure in the target logic that behaves similarly. Thus, through a grammatical translation, the connectives of the source logic are generally preserved in the target logic. Koslow [Kos15, p. 48] also allows that a connective from one logic “persists” in if there is a homomorphism from to .
According to Mossakowski et al. [MDT09], grammatical translations are too demanding for the task, as many useful and important translations are non-grammatical (e.g. the standard modal translation). For them, instead of seeking to preserve the structure of the formulas, it would be better to preserve the proof-theoretic behaviour of the connectives and to treat the connectives only as regards this behaviour (ibid, p. 100). In this paper, some proof-theoretic conditions on the connectives are listed, e.g. for conjuntction the condition is iff and . This formulation may lead one to think that here shall be a logical constant, and not possibly a formula (think of , where ); naturally in the first case, the whole proposal would make no sense. In table 1 we reformulate the conditions to reflect their proposal more clearly, where is an arbitrary formula that stands for the connective .
|falsum||, for every|
Definition 3.11 (presence of a proof-theoretic connective).
A proof-theoretic connective is present in a logic if it is possible to define the corresponding operations on sentences satisfying the conditions given in table 1.
We shall now investigate this idea in detail and argue that, as it is, the preservation of connectives would require mappings stricter than conservative translations otherwise the notion of the “presence” of a connective must be relaxed.
Drawbacks on the preservation of proof-theoretic connectives A translation transports a given -connective if its presence in implies its presence in , the converse implication is called reflection [MDT09, p. 100]. It is claimed (ibid) that if a mapping is conservative and surjective, then all proof theoretic connectives of are transported to and all proof-theoretic connectives present in are reflected in . However, this claim must be taken with a grain of salt, let us see why.
Let be a logic having a proof-theoretic conjunction according with the table 1 above and suppose there is a surjective conservative mapping . For -formulas , let be a set of -formulas with and . Then it holds that
( and ) iff iff .
Thus, would have proof-theoretic conjunction. The grain of salt is that, once no structural restriction is imposed upon , it is not necessary that be constructed out of and . In this case, it seems at least unnatural to say that is an operation on the sentences and .
Therefore, we must relax what it means for a connective to be present in a logic. One has to say that e.g. the proof-theoretic conjunction is present in a logic if, for all formulas and set of formulas , there is a formula such that ( and ) iff . A similar reformulation should be given for the other connectives. In this case, though, whenever it holds that and , then any -theorem in the place of serves to satisfy this condition for conjunction.
For example, take a Tarskian logic defined on the signature , where are propositional variables and the constant for logical truth. Then has proof-theoretic conjunction since and holds iff . This is probably unproblematic and a consequence of the meaning of . Nevertheless, for some cases this approach to the presence of connectives has some downsides. For example, restrict to the signature . Then has the proof-theoretic conditional, since it holds that
iff , iff , iff and iff .
But if the signature were incremented by another variable , then the resulting system would no longer have a proof-theoretic conditional, since for no it would hold that iff . This volatility of the presence of proof-theoretic connectives is unreasonable.
Recapitulating, the idea of this approach is that one shall define the mappings so as to preserve the proof-theoretic connectives, instead of requiring the mappings themselves to preserve the structure of the formulas. But if the mappings do not respect the structure of the formulas, what shall be called the presence of a connective, must also be relaxed.
Besides the inconvenients mentioned above, this proposal would be too restrictive in some cases. For example, Statman’s translation [Sta79] of into its implicational fragment shows how can one “express” (in some sense of the term) conjunctions using only implicational formulas; recent works have generalized this result so that any logic having a certain natural deduction formulation and having the sub-formula principle is translatable into the implicational fragment of minimal logic [Hae15].
Anyway, it must be borne in mind that to give a good and general definition of when a connective or operator is generally preserved is a difficult and spinous topic. Below we give another proposal, which is at the same time weaker (the translation mentioned above would enter) and stronger (requires structure-attentive mappings).
Let us now consider the structure-attentive translations and think on the minimum conditions on the preservation of the structure of formulas that would allow for a reasonable and general notion of preservation of connectives.
General-recursive translations: allowing context-sensitivity in a general preservation of connectives The criterion of compositionality given above a priori seems a reasonable condition for the preservation of connectives through translations. Notice that in the criterion the function that translates is the same that translates the sub-formulas . From this comes the compositionality: a translation of a formula is obtained through the same translation of its sub-formulas.
Thinking about the issue of translating a connective, it is also reasonable that the translation be sensitive to the context where the connective is inserted. This is the case in the translation in [DG00], where , but . Therefore, distinguishes between translating -formula and -formula, and this is done through the help of an auxiliary translation (see complete definition in section 3.27). Thus, there are translations between some logics where the mappings must be context-sensitive, so as to convey the proper meaning of some source connectives in the target logic.
There are also those cases where the connectives can be dealt context-independently but auxiliary translations are needed anyway. The standard translation of modal logic to , besides some parameters, needs auxiliary translations for each formula of modal degree e.g. as but .
For the sake of simplicity, we will restrict our notion of context-sensitivity to whether or not the connective to be translated is in the scope of an unary operator. When the translation of a -ary connective is sensitive as to whether it is on the scope of an unary , a simple solution is to treat as a composite -ary connective to be translated. With the aim of capturing these cases, let us consider a sufficiently general kind of translation.
French in [Fre10] presents a concept of recursively interdependent translation that includes non-compositional translations that are still defined recursively through the formation of formulas. A generalization of his concept will be employed here, since the original has an unmotivated restriction allowing only unary auxiliary mappings. The generalization allows auxiliary mappings of any arity and also has a simpler notation. Let and be logics,
Definition 3.12 (General-Recursive).
Let be auxiliary mappings of any arity defined inductively on -formulas. A translation from to is general-recursive if, for every -ary connective and formulas